If and are homotopy equivalent, then is isomorphic to More specifically, if is a homotopy, then and are isomorphisms, and .
Proof:
Let be a point-wise continuous path of -homomorphisms connecting and . Extend this path to a pointwise continuous -homomorphisms for each .
For every projection the path is continuous (pointwise continuous) and so . This shows that The map sends projections to projections, and these two projections are homotopic, so by shifting up to matrices, we get that they are K0 equivalent???